MathDB

p1. For positive real numbers

x, y

, the operation

\otimes

is given by

x \otimes y =\sqrt{x^2 - y}

and the operation

\oplus

is given by

x \oplus y =\sqrt{x^2 + y}

. Compute

(((5\otimes 4)\oplus 3)\otimes2)\oplus 1

.
p2. Janabel is cutting up a pizza for a party. She knows there will either be

4

5

, or

6

people at the party including herself, but can’t remember which. What is the least number of slices Janabel can cut her pizza to guarantee that everyone at the party will be able to eat an equal number of slices?
p3. If the numerator of a certain fraction is added to the numerator and the denominator, the result is

\frac{20}{19}

. What is the fraction?
p4. Let trapezoid

ABCD

be such that

AB \parallel CD

. Additionally,

AC = AD = 5

CD = 6

, and

AB = 3

. Find

BC

.
p5. AtMerrick’s Ice Cream Parlor, customers can order one of three flavors of ice cream and can have their ice cream in either a cup or a cone. Additionally, customers can choose any combination of the following three toppings: sprinkles, fudge, and cherries. How many ways are there to buy ice cream?
p6. Find the minimum possible value of the expression

|x+1|+|x-4|+|x-6|

.
p7. How many

3

digit numbers have an even number of even digits?
p8. Given that the number

1a99b67

is divisible by

7

9

, and

11

, what are

a

and

b

? Express your answer as an ordered pair.
p9. Let

O

be the center of a quarter circle with radius

1

and arc

AB

be the quarter of the circle’s circumference. Let

M

N

be the midpoints of

AO

and

BO

, respectively. Let

X

be the intersection of

AN

and

BM

. Find the area of the region enclosed by arc

AB

AX

BX

.
p10. Each square of a

5

-by-

1

grid of squares is labeled with a digit between

0

and

9

, inclusive, such that the sum of the numbers on any two adjacent squares is divisible by

3

. How many such labelings are possible if each digit can be used more than once?
p11. A two-digit number has the property that the difference between the number and the sum of its digits is divisible by the units digit. If the tens digit is

5

, how many different possible values of the units digit are there?
p12. There are

2019

red balls and

2019

white balls in a jar. One ball is drawn and replaced with a ball of the other color. The jar is then shaken and one ball is chosen. What is the probability that this ball is red?
p13. Let

ABCD

be a square with side length

2

. Let

\ell

denote the line perpendicular to diagonal

AC

through point

C

, and let

E

and

F

be themidpoints of segments

BC

and

CD

, respectively. Let lines

AE

and

AF

meet

\ell

at points

X

and

Y

, respectively. Compute the area of

\vartriangle AXY

.
p14. Express

\sqrt{21-6\sqrt6}+\sqrt{21+6\sqrt6}

in simplest radical form.
p15. Let

\vartriangle ABC

be an equilateral triangle with side length two. Let

D

and

E

be on

AB

and

AC

respectively such that

\angle ABE =\angle ACD = 15^o

. Find the length of

DE

.
p16.

2018

ants walk on a line that is

1

inch long. At integer time

t

seconds, the ant with label

1 \le t \le 2018

enters on the left side of the line and walks among the line at a speed of

\frac{1}{t}

inches per second, until it reaches the right end and walks off. Determine the number of ants on the line when

t = 2019

seconds.
p17. Determine the number of ordered tuples

(a_1,a_2,... ,a_5)

of positive integers that satisfy

a_1 \le a_2 \le ... \le a_5 \le 5

.
p18. Find the sum of all positive integer values of

k

for which the equation

\gcd (n^2 -n -2019,n +1) = k

has a positive integer solution for

n

.
p19. Let

a_0 = 2

b_0 = 1

, and for

n \ge 0

, let

a_{n+1} = 2a_n +b_n +1,

b_{n+1} = a_n +2b_n +1.

Find the remainder when

a_{2019}

is divided by

100

.
p20. In

\vartriangle ABC

, let

AD

be the angle bisector of

\angle BAC

such that

D

is on segment

BC

. Let

T

be the intersection of ray

\overrightarrow{CB}

and the line tangent to the circumcircle of

\vartriangle ABC

A

. Given that

BD = 2

and

TC = 10

, find the length of

AT

.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement